Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 24 / 3 / ii

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- Solution ii

Solution

Choose

b = (a d - 1) / N \in Z

, and take the matrix

γ = (a N b d) \in S L_{2} (Z) .

At

w = τ - d / N

its denominator is

Nw + d = N τ

, while its numerator is

a w + b = a τ - 1/ N

. Consequently

γ w = \frac{a}{N} - \frac{1}{N ^{2} τ} .

Applying the weight-

k

transformation law of the cusp form at

w \in H

proves the rational-cusp inversion of a level-one cusp form:

f (\frac{a}{N} - \frac{1}{N ^{2} τ}) = (N τ)^{k} f (τ - \frac{d}{N}) .

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